Determining the countability of a set of function.

**auitto** · October 12th 2010, 09:28 AM

Determine whether or not the set D of all functions $f:\mathbb{Z}_+\to\mathbb{Z}_+{$ is countable. Justify your answer.

If anyone could give me some hints I could use to get started on this question I would really appreciate it. I have been trying to use the fact that D is a subset of the power set of $\mathbb{Z}_+\times\mathbb{Z}_+$ but is have not really been able to get anywhere with that.

**Plato** · October 12th 2010, 09:55 AM

The set of all functions $f:\mathbb{Z}^+\to \{0,1\}=2^{\mathbb{Z}^+ }$ which in equipotent to $\mathscr{P}\left(\mathbb{Z}^+ \right)$ , the power set of $\mathbb{Z}^+$ .

Is it not true that $\left| {\mathbb{Z}^ + } \right| < \left| {\mathscr{P}\left( {\mathbb{Z}^ + } \right)} \right|?$

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